series convergence/divergence tests (calc II)

This is a discussion on series convergence/divergence tests (calc II) within the A Brief History of Cprogramming.com forums, part of the Community Boards category; Alright, so tell me if this is bunk. I have gone through every problem in my book. I can look ...

What do you mean? Knowing integral convergence test, nth term test, alternating series test, and etc...? All these theorems are there mainly to take in to account most of the general equations you'll run in to, so they're good to know.

Also, they're the formal way of proving that the series converges/diverges. Just "knowing" that a series conv/div isn't good enough in academia.

Alright, so tell me if this is bunk. I have gone through every problem in my book. I can look at the series for around 30 seconds or less, and decide whether it converges or diverges.

You've misunderstood the purpose of the problems in your book. Their purpose is to help you understand the theory, not to learn by heart to avoid the theory.
In real life/other courses you won't have any benefits from being able to look at a series and tell if it converges. If you learn all the 'stupid tests' though, you'll gain useful knowledge of calculus.

There are 10 types of people in this world, those who cringed when reading the beginning of this sentence and those who salivated to how superior they are for understanding something as simple as binary.

Okay, so both of those fairly obviously diverge. You probably couldn't prove it easily, though, unless you had the right tools available. And if somebody sat there convinced that the sum of the reciprocals of primes converged, how would you convince them?

In math, you can't get away with going by your gut feeling, because at one point in time or another, you'll be wrong. For example, the sum of

1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + - ...

is log(2). Well all I'm going to do is move a few terms around. What's the value of the sum,

There are 10 types of people in this world, those who cringed when reading the beginning of this sentence and those who salivated to how superior they are for understanding something as simple as binary.

There are 10 types of people in this world, those who cringed when reading the beginning of this sentence and those who salivated to how superior they are for understanding something as simple as binary.

This argument applies for the rest of the parenthesized groups. So the sum for each parenthesized group is less than 9/10 the sum of the previous parenthesized group. From there we know that the overall sum is less than that of a geometric series with ratio 9/10, whose first term is 1+1/2+...+1/8.

There are 10 types of people in this world, those who cringed when reading the beginning of this sentence and those who salivated to how superior they are for understanding something as simple as binary.

If you're trying to invent an algorithm that approximates something, you'll want to know whether it approximates that thing well, and how quickly it can come up with that approximation. Your approximation method might only work for certain inputs, so you'd need to know which inputs these work on. Or it might only work quickly for certain inputs. With certain restrictions of input, you can have a guarantee of how long it takes to get a certain degree of approximation.

There are 10 types of people in this world, those who cringed when reading the beginning of this sentence and those who salivated to how superior they are for understanding something as simple as binary.